Topics in Computer Aided Geometric Design, Erice May 12-19, 1990 - - PowerPoint PPT Presentation

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Topics in Computer Aided Geometric Design, Erice May 12-19, 1990 - - PowerPoint PPT Presentation

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Topics in Computer Aided Geometric Design, Erice May 12-19, 1990 Smooth Simplex Splines for the Powell-Sabin 12-split Tom Lyche Centre of Mathematics for Applications, Department of Mathematics, University of Oslo September 28, 2013

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Topics in Computer Aided Geometric Design, Erice May 12-19, 1990

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Smooth Simplex Splines for the Powell-Sabin 12-split

Tom Lyche

Centre of Mathematics for Applications, Department of Mathematics, University of Oslo

September 28, 2013

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Outline

Simplex splines A quadratic simplex spline basis for PS12 on one triangle Higher degree splines

  • n the 12-split
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SLIDE 4

PART I: simplex splines

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Schoenberg’s View of the Bivariate B-Spline

In a letter from Iso Schoenberg to Phillip Davis from 1965: “A sketch of the spline function z = M(x, y; z0, z1, z2, z3, z4)”

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SLIDE 6

Simplex spline definitions

◮ geometric ◮ variational ◮ recurrence relation

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SLIDE 7

Recurrence Relation

  • [x0,...,xn]

f :=

  • ∆n f (v0x0 + v1x1 + · · · + vnxn)dv1 · · · dvn
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SLIDE 8

Bivariate simplex spline properties

◮ Let X be a collection of d + 3 points x1, . . . , xd+3 in R2 ◮ A simplex spline S = S[X] : R2 → R with knots X is a

nonnegative piecewise polynomial

◮ the degree is d ◮ the support is the convex hull of X ◮ the knotlines are the edges in the complete graph of X ◮ a knot line has multiplicity m if it contains m + 1 of the

points in X

◮ S ∈ C d−m across a knotline of multiplicity m

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SLIDE 9

Some simplex spline spaces

◮ Triangulate a slab, de Boor, 1976. ◮ Complete Configurations, Hakopian[1981], Dahmen,

Michelli[1983],

◮ Pull apart, Dahmen, Micchelli[1982], H¨

  • llig[1982], Dahmen,

Micchelli,Seidel[Erice 1990],

◮ Delaunay configurations, Neamtu[2000-2007]

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”There is no clever way to implement the recurrence relation

  • nce the standard recipe for constructing spaces of simplex

spline functions has been followed” Tom Grandine, 1987

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What should be the space of Simplex splines on a triangulation?

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PART II: A Simplex spline basis for PS12

  • n one triangle

Cohen, E., T. Lyche, and R. F. Riesenfeld, A B-spline-like basis for the Powell-Sabin 12-split based on simplex splines, Math. Comp., 82(2013), 1667-1707

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The PS12-split (Powell,Sabin 1977)

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The PS12-split

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The PS12 spline space

S1

2(

) = {f ∈ C 1( ) : f|∆i ∈ Π2, i = 1, . . . , 12} dim(S1

2(∆PS12)) = 12

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SLIDE 16

Computing with PS12

◮ Bernstein-B´

ezier methods,

◮ FEM nodal basis, (Oswald) ◮ minimal determining set (Alfeld, Schumaker, Sorokina) ◮ subdivision (Dyn, Lyche, Davydov, Yeo) ◮ quadratic S(implex) - splines (Cohen, Lyche, Riesenfeld)

A Hermite Sub division Scheme for PS-12 3

Initialization

  • 1. subdivision

step

Fig. 2. Sub dividing the PS-12 split elemen t. A circle around a v ertex means that b
  • th
the function v alue and the gradien t are kno wn at that v ertex. 2.1. Initializati
  • n.
The rst step in the computation
  • f
suc h an elemen t in v
  • lv
es the computation
  • f
its v alue and gradien t at the midp
  • in
ts a; b; c
  • f
the triangle T , (see, Fig. 1). Here w e use the form ula f b = (f A + f C )=2
  • (rf
A
  • rf
C )
  • (A
  • C
)=8 for the function v alue at the midp
  • in
t b
  • f
AC . F
  • r
the gradien t w e rst compute the directional deriv ativ e in the direction AC at b (A
  • C
)
  • rf
b = 2(f A
  • f
C )
  • (rf
A + rf C )
  • (A
  • C
)=2: Com bining this v alue with the giv en v alue
  • f
the cross-deriv ativ e at b, w e can calculate rf b . F
  • r
the
  • ther
midp
  • in
ts w e use similar form ulae. These form ulae are
  • btained
from the
  • bserv
ation that along eac h side
  • f
T the PS-12 split elemen t is a piecewise quadratic C 1
  • spline
with a knot at the midp
  • in
t. 2.2. The General Sub division Step. F
  • r
the rst sub division step (see Fig. 1) w e use the follo wing form ulae: f e = (f b + f C )=2
  • (rf
b
  • rf
C )
  • (b
  • C
)=8 f g = (f a + f C )=2
  • (rf
a
  • rf
C )
  • (a
  • C
)=8 f d = (f b + f a )=2
  • (rf
b
  • rf
a )
  • (b
  • a)=8
rf e = (rf b + rf C )=2 rf g = (rf a + rf C )=2 (a
  • b)
  • rf
d = 2(f a
  • f
b )
  • (rf
a + rf b )
  • (a
  • b)=2
(C
  • d)
  • rf
d = 2(f C
  • f
d )
  • rf
C
  • (C
  • d):
(1) >F rom the last t w
  • v
alues w e can solv e for rf d . Similar form ulae are used for the t w
  • ther
corner triangles Abc and B ca and w e
  • btain
the v alues and gradien ts at lo cations sho wn to the righ t in Fig. 2. This pro cess can no w b e con tin ued for as man y lev els
  • f
renemen t as desired. Fig. 3 displa ys a PS-12 split elemen t
  • btained
from random initial data. The implemen tation w as done using Mathematica.
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3 corner S-splines for the quadratic case; support 1/4

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6 half support S-splines for the quadratic case

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3 trapezoidal support S-splines for the quadratic case

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Properties

◮ Local linear independence, ◮ nonnegative partition of unity, ◮ stable recurrence relations, ◮ fast pyramidal evaluation algorithms, ◮ differentiation formula, ◮ Lp stable basis, ◮ subdivision algorithms of Oslo- and Lane,Riesenfeld type, ◮ quadratic convergence of control mesh, ◮ well conditioned collocation matrices for Lagrange and

Hermite interpolation,

◮ explicit dual functionals, ◮ dual polynomials and Marsden-like identity.

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Marsden-like identity

Univariate quadratic: (1 − yx)2 =

j Bj,2(x)(1 − ytj+1)(1 − ytj+2)

Quadratic S-splines: x ∈ ∆, y ∈ R2 (1 − yTx)2 =

12

  • j=1

Sj,2(x)(1 − yTp∗

j,1)(1 − yTp∗ j,2).

1 2 3 5 7 6 8 4 9 10

1 2 3 4 5 6 7 8 9 10 11 12

[p∗

1,1, . . . , p∗ 12,1] := [p1, p1, p4, p4, p2, p2, p5, p5, p3, p3, p6, p6],

[p∗

1,2, . . . , p∗ 12,2] := [p1, p4, p10, p2, p2, p5, p10, p3, p3, p6, p10, p1], ◮ 1 = 12 j=1 Sj,2(x),

x ∈ ∆,

◮ x = 12 j=1 Sj,2(x)mj,

x ∈ ∆, mj := (p∗

j,1 + p∗ j,2)/2.

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domain- and control mesh

The control points are at a distance O(h2) from the surface, where h is the longest side of the triangle..

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Dual functionals

Univariate quadratic: λjf := 2f (tj+3/2) − 1 2f (tj+1) − 1 2f (tj+2), λiBj,2 = δij Quadratic S-splines: λjf := 2f (mj) − 1 2f (p∗

j,1) − 1

2f (p∗

j,2)

λiSj,2 = δij

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C 1 smoothness is controlled as in the polynomial B´ ezier case.

1 2 12 3 4 7 11 5 6 8 9 10 1 2 12 3 4 7 11 5 6 8 9 10

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PART III: Higher degree splines on the 12-split

Joint work with Georg Muntingh

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Smooth splines on triangulation

◮ Consider a general triangulation in the plane ◮ consider a subdivided triangulation with the 12-split on each

triangle and a piecewise polynomial of degree d on this triangulation

◮ d = 2 necessary and sufficient for C 1 ◮ d = 5 necessary and sufficient for C 2

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Dimensions of Sr

d( )

Sr

d(

) = {f ∈ C r( ) : f|∆i ∈ Πd, i = 1, . . . , 12}

Theorem

For any integers d, r with d ≥ 0 and d ≥ r ≥ −1 dim Sr

d(

) = 1 2(r + 1)(r + 2) + 9 2(d − r)(d − r + 1) + 3 2(d − 2r − 1)(d − 2r)+ +

d−r

  • j=1

(r − 2j + 1)+, (1) where z+ := max{0, z} for any real z.

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Proof

One cell and three flaps. Use cell dimension formula in Lai-Schumaker book.

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Dimensions of Sr

d( )

d/r −1 1 2 3 4 5 6 7 8 9 10 11 12 1 1 36 10 3 2 72 31 12 6 3 120 64 30 16 10 4 180 109 60 34 21 15 5 252 166 102 61 39 27 21 6 336 235 156 100 66 46 34 28 7 432 316 222 151 102 73 54 42 36 8 540 409 300 214 150 109 81 63 51 45 9 660 514 390 289 210 154 117 91 73 61 55 10 792 631 492 376 282 211 162 127 102 84 72 66 11 936 760 606 475 366 280 216 172 138 114 96 84 78

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An interesting family on

◮ For any positive integer n consider on

the spline space S2n−1

3n−1(

) of splines of smoothness 2n − 1 and degree 3n − 1 .

◮ n = 1: C 1 quadratics ◮ n = 2: C 3 quintics ◮ n = 3: C 5 octic ◮ dim S2n−1 3n−1(

) = 15

2 n2 + 9 2n.

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Hermite degrees of freedom S3

5( )

◮ dim S3 5(

) = 39

◮ 10 derivatives at 3 corners ◮ 3 first order cross boundary derivatives ◮ 6 second order cross boundary derivatives ◮ Connects to neighboring triangles with smoothness C 2.

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Spline space on triangulation of smoothness C n

◮ Consider a triangulation in the plane ◮ use S2n−1 3n−1(

) on each triangle

◮ get a global spline space of smoothness C n ◮ for n = 2 we get a C 2 spline space of dimension 10V + 3E ◮ for n ≥ 1 we get a C n spline space of dimension

n(2n + 1)V + 1 2n(n + 1)E

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Simplex spline basis for S3

5( )

6 1 1 1 5 2 4 2 1 1 3 2 2 1

3 6 6 6

4 2 1 1 4 2 2 1 1 1 1 2 2 1 1 2 2 2

6 3 6 3

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Simplex spline basis for S3

5( )

0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.2 0.4

0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.1 0.2 0.3 0.4

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Simplex spline basis for S3

5( );

◮ nonnegative partition of unity ◮ globally linearly independent ◮ can be computed recursively ◮ reduces to univariate quintic B-splines on boundary ◮ not locally linearly independent ◮ no simplex spline basis for for S3 5(

) that is locally linearly independent

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Thank you!

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